Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#11048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,7)(8,9,12,17,13,16,25,31)(10,14,20,28,19,27,35,38)(11,21,22,32,23,15,24,30)(18,26,34,37,33,29,36,39);(4,5,6,7)(8,10)(9,14)(11,18)(12,20)(13,19)(15,29)(16,27)(17,28)(21,26)(22,34)(23,33)(24,36)(25,35)(30,39)(31,38)(32,37);(8,11,13,23)(9,15,16,21)(10,18,19,33)(12,24,25,22)(14,26,27,29)(17,32,31,30)(20,36,35,34)(28,39,38,37)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4036);